"Greg Egan - Foundations 1 - Special Relativity" - читать интересную книгу автора (Egan Greg)

(x,y) = (OQ, OR)
= (OT, OU) + (тАУQT, UR)


Making use of Eqn (2c):


g[(x,y),(cos A, sin A)] = g[(OT, OU),(cos A, sin A)] + g[(тАУQT, UR),(cos A, sin A)]


= OS + (тАУQT cos A + UR sin A)
= OS + (тАУPS sin A cos A + PS cos A sin A)
= OS


OS = g[(x,y),(cos A, sin A)] (6)


What's this vector (cos A, sin A)? It points in the same direction as the vector
OG, but from what we know about sine and cosine:


тИЪ((cos A)2 + (sin A)2)
|(cos A, sin A)| =
= 1

A vector like this, with a magnitude of one, is known as a unit vector. Eqn (6)
tells us that to calculate the length of the projection of (x,y) onto OG, we just apply the
 Egan: "Foundations 1"/p.8


metric function to (x,y) and the unit vector in the direction of OG.
What if we don't know the angle A, but only the coordinates of G, (u,w)? We
can still compute the unit vector in this direction, just by dividing (u,w) by its own
length, automatically re-scaling it to a length of one.


(cos A, sin A) = (u,w) / |(u,w)|
OS = g[(x,y),(cos A, sin A)]
= g[(x,y),(u,w)/|(u,w)|]
= g[(x,y),(u,w)] / |(u,w)| (7)


Here we've made use of Eqn (2b) to shift the factor 1/|(u,w)| outside the metric.
Equating the two formulas for OS from Eqns (5) & (7) gives:


|(x,y)| cos B = g[(x,y),(u,w)] / |(u,w)|
cos B = g[(x,y),(u,w)] / (|(x,y)||(u,w)|) (8a)
g[(x,y),(u,w)] = |(x,y)||(u,w)| cos B (8b)